Comparison and analysis of Speed Control of DC Motor Using Advanced Traditional LQR Tuning Controller

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(1)IJRIT International Journal of Research in Information Technology, Volume 2, Issue 5, May 2014, Pg: 342-348. International Journal of Research in Information Technology (IJRIT). www.ijrit.com. ISSN 2001-5569. Comparison and analysis of Speed Control of DC Motor Using Advanced Traditional LQR Tuning Controller Vivek Shrivastva1, Rameshwar Singh2 1. M.Tech Student, Deptt. of Electrical,NITM,RGPV , Gwalior, India. [email protected], 2. Assistant professor, Deptt of Deptt. Of Electrical, NITM, Gwalior, India [email protected]. Abstract This paper presented a study of tuning of Linear Quadratic Regulator (LQR) Controller for speed control of DC motor. DC motor is widely used in industries even if its maintenance cost is higher than the induction motor. Speed control of DC motor is attracted considerable research and several methods are evolved. The LQR controller is the very commonly used compensating controller. Linear Quadratic Regulator (LQR), the tuning method was more efficient in improving the step response characteristics such as, reducing the rise time, settling time and maximum overshoot in speed control of DC motor. Linear quadratic regulator method gives the better performance compare the are various tuning techniques. The results obtained are compared Linear Quadratic Regulator (LQR) Controller with Proportional-Integral-Derivative Controller (PID), Fuzzy Logic Controller (FLC) and Fractional Order PID Controller (FOPID). Keywords-Proportional-Integral Derivative Controller, FOPID, FLC, Linear quadratic regulator, DC motor.. I. Introduction DC motor is a power actuator which converts electrical energy into mechanical energy. DC motor is used in application where wide speed ranges are required. The greatest advantage of dc motors may be speed control. The term speed control stand for intentional speed variation carried out manually or automatically. DC motors are most suitable for wide range speed control and are therefore used in many adjustable speed drives Because of their high reliabilities, flexibilities and low costs, DC motors are widely used in industrial applications, robot manipulators and home appliances where speed Control of motor is required. PID controllers are commonly used for motor control applications because of their simple structures and intuitionally comprehensible control algorithms. Controller parameters are generally tuned using hand-tuning or Ziegler-Nichols frequency response method. Both of these methods have successful results but long time and effort are required to obtain a satisfactory system response. Two main problems encountered in motor control are the time-varying nature of motor Parameters under operating conditions and existence of noise in system loop [1], several approaches have been documented in literatures for determining the PID parameters of such controllers which is first found by Ziegler- Nichols tuning [2]. Genetic Algorithm, [3], Ziegler-Nichols presented a tuning formula [4 ]. The other type of control methods can be developed such as Linear- Quadratic Regulator (LQR) optimal control, linear quadratic regulator design technique is well known in modern optimal control theory and has been widely used in many applications. It has a very nice robustness property. It is attractive property appeals to the practicing engineers. Thus, the linear quadratic regulator theory has received considerable attention since 1950s [5].The liner quadratic regulator technique seeks to find the optimal controller that minimizes a given cost function (performance index). This cost function is parameterized by two matrices, Q and R, that weight the state vector and the system input respectively. These weighting matrices Vivek Shrivastva, IJRIT. 342.

(2) IJRIT International Journal of Research in Information Technology, Volume 2, Issue 5, May 2014, Pg: 342-348. regulate the penalties on the excursion of state variables and control signal. One practical method is to Q and R to be diagonal matrix. The value of the elements in Q and R is related to its contribution to the cost function. To find the control law, Algebraic Riccati Equation (ARE) is first solved, and an optimal feedback gain matrix, which will lead to optimal results evaluating from the defined cost function is obtained [6].. II . Modeling of Dc Motor In normal conditions and without controller, the DC motor does not have an acceptable performance, this fact will be shown in later sections. Because of analysis a DC motor and show its performance; in this section it is described how to develop a linear model for a DC motor, and how to analyze the model under Matlab/Simulink. we need a conceptual realization of a DC [12]. DC motor shown in Fig. 1 is the one of most common motors which used in industrial motion control systems.. Fig 1 DC motor [7]. a) Physical System Electric circuit of the armature and the free body diagram of the rotor of a DC motor, are shown in Figure 2. The rotor and the shaft are assumed to be rigid. Consider the following values for the physical parameters show in the table 1. Table 1 Parameter values of DC motor [8]. The input is the armature voltage, Va, (driven by a voltage source). Measured variables are the angular velocity of the shaft (ω) in radians per second, and the shaft angle (θ) in radians.. Vivek Shrivastva, IJRIT. 343.

(3) IJRIT International Journal of Research in Information Technology, Volume 2, Issue 5, May 2014, Pg: 342-348. Fig 2 Direct Current Motor Model. b) System Equations A linear model of a simple DC motor consists of an electrical equation and mechanical equation. Using Kirchhoff’s Voltage Law (KVL) and Newton’s second law, the following equation is obtained: .

(4) … . . . . 1 .

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(6) … … … … … … … 2 . Assuming the above equations, the steady state representation can be obtained as, fig 3 show the dc motor armature control system. di R # dt " L dω " K ( dt ! J. K ' # / L " i , - . 01 3 4V 6 3 Bm " ω 0 J ! 8 40 16 9 : ……………(4)

(7). c) Transfer Function The transfer function of the motor using the state space model by formula G(s)= C (s I - A)-1 B + D in the equation (3) and (4) and obtain the equation 5 .. Fig. 3- Block Diagram of DC Motor

(8) <. ;< … . 5 < ∗ > ? @ ? ∗ > ∗ ∗ ∗ . Vivek Shrivastva, IJRIT. 344.

(9) IJRIT International Journal of Research in Information Technology, Volume 2, Issue 5, May 2014, Pg: 342-348. III. Linear-Quadratic Regulator (LQR) Controller LQR is a method in modern control theory that used state-space approach to analyze such a system. Using state space methods it is relatively simple to work with Multi- Input Multi-Output (MIMO) system [9]. LQR is a method in modern control theory that used state-space approach to analyse such a system. Using state space methods it is relatively simple to work with Multi- Input Multi-Output system. Linear quadratic regulator design technique is well known in modern optimal control theory and has been widely used in many applications, Linear-Quadratic Regulator (LQR) optimal control problems have been widely investigated in the literature. The performance measure is a quadratic function composed of state vector and control input. If the linear time-invariant system is controllable, the optimal control law will be obtained via solving the algebraic Ricci equation optimal control. The function of Linear Quadratic Regulator (LQR) is to minimize the deviation of the speed of the motor. The speed of the motor is specifying that will be the input voltage of the motor and the output will be compare with the input. In general, the system model can be written in state space equation as follows: .B CD E … . 6 A is the state matrix of order G × G B is the control matrix of order G × . Also, the pair (A, B) is assumed to be such that the system is controllable. The linear quadratic regulator controller design is a method of reducing the performance index to a minimize value. The minimization of it is just the means to the end of achieving acceptable performance of the system. For the design of a linear quadratic regulator controller, the performance index (J) is given by: L. I D JB E K … N M. Where Q is symmetric positive semi-definite state weighting matrix of order G × G and R is symmetric positive definite control weighting matrix of order G × The choice of the element Q and R allows the relative weighting of individual state variables and individual control inputs as well as relative weighting state vector and control vector against each other. The weighting matrices Q and R are important components of an LQR optimization process. The compositions of Q and R elements have great influences of system performance. The designer is free to choose the matrices Q and R, but the selection of matrices Q and R is normally based on an iterative procedure using experience and physical understanding of the problems involved. Commonly, a trial and error method has been used to construct the matrices Q and R elements. This method is very simple and very familiar in linear quadratic regulator application. However, it takes long time to choose the best values for matrices Q and R. The number of matrices Q and R elements are dependent on the number of state variable (n) and the number of input variable (m), respectively. The diagonal-off elements of these matrices are zero for simplicity. If diagonal matrices are selected, the quadratic performance index is simply a weighted integral of the squared error of the states and inputs. The term in the brackets in equation (8) above are called quadratic forms and are quite common in matrix algebra. Also, the performance index will always be a scalar quantity, whatever the size of Q and R matrices .The conventional linear quadratic regulator problem is to find the optimal control input law u* that minimizes the performance index under the constraints of Q and R matrices[10].The closed loop of dc motor with Linear quadratic regulator show in the fig 1, The closed loop optimal control law is defined as: ∗K D … . 8. Where K is the optimal feedback gain matrix, and determines the proper placement of closed loop poles to minimize the performance index in equation (7). The feedback gain matrix K depends on the matrices A, B, Q, and R. There are two main equations which have to be calculated to achieve the feedback gain matrix K. Where P is a symmetric and positive definite matrix obtained by solution of the ARE is defined as: C P PC – P R/ P J 0 …(9). Then the feedback gain matrix K is given by: Substituting the above equation (8) into Equation (6) gives:. . T . K = R/ P … . 10 CS D C D …(11). If the Eigen values of the matrix (A-BK) have negative real parts, such a positive definite solution always exits [11]. IV Analysis of Result All the conventional methods of controller tuning lead to a large settling time, overshoot, rise time and steady state error of the controlled system. Hence a Soft computing techniques is introduces into the control loop.LQR, based tuning methods have proved their excellence in giving better results by improving the steady state characteristics. Vivek Shrivastva, IJRIT. 345.

(10) IJRIT International Journal of Research in Information Technology, Volume 2, Issue 5, May 2014, Pg: 342-348. and performance indices. Simulations were carried out using MATLAB 7.0.1 on a Pentium IV processor, 2.8 GHz. with 1 GB RAM. ..(12) the final transfer function of DC motor becomes UV . WV 1.25 … . 13 XY V 0.002? @ 0.05016? 1.5665. CaseI: the data of dc motor show in the table 1 and transfer function of dc motor equation 12 and 13 used as a system and find out the response of the system applying the step function as an input. And the tuning of different point such as the LQR parameter Q and R ,so better result show in the table 2 and The output response shown in Fig. 4,Fig 5 and Fig 6. Table 2 best result of LQR controller for case I. Fig 4 Speed Response of LQR Controller Case I. Vivek Shrivastva, IJRIT. Fig 5 speed response of LQR Controller case I. 346.

(11) IJRIT International Journal of Research in Information Technology, Volume 2, Issue 5, May 2014, Pg: 342-348. Fig 6 speed response of Linear-Quadratic Regulator Controller Case I. Case II. Comparison of Proportional-Integral-Derivative Controller (PID), Fractional Order PID Controller [FOPID], Fuzzy Logic Controller (FLC) and LQR for speed control of dc motor ,the comparison of different tuning controller, PID,FOPID ,FLC with LQR controller ,minimum rise time and minimum setting time Achieved In LQR Controller ,the best result show in the table 3 and fig 7. Table 3 comparative analysis of different controllers. Fig 7 speed response of Linear-Quadratic Regulator Controller Case II. IV.CONCLUSON In this paper compare of PID, FLC.FOPID techniques to control the speed control of dc motor. Our aim is to improve the dynamic performance of the system output like settling time, rise time and maximum overshoot. Comparing our proposed controller to LQR based controller, we observe overshoot and settling time and rise time and final value are improved in proposed controller. Vivek Shrivastva, IJRIT. 347.

(12) IJRIT International Journal of Research in Information Technology, Volume 2, Issue 5, May 2014, Pg: 342-348. REFERENCES [1] Husain Ahmed1, Dr. Gagan Singh2, Vikash Bhardwaj3 , Saket Saurav4, Shubham Agarwal, “Controlling of D.C. Motor using Fuzzy Logic Controller”, Conference on Advances in Communication and Control Systems 2013 (CAC2S 2013). [2] J. C. Basilio and S. R. Matos, “Design of PI and PID Controllers With Transient Performance Specification”, IEEE Trans. Education, vol. 45, Issue No. 4, 2002, pp. 364-370. [3] Megha Jaiswal , Mohna Phadnis (H.O.D. EX), “Speed Control of DC Motor Using. Genetic Algorithm Based. PID Controller”, International Journal of Advanced Research in Computer Science and Software Engineering, Volume 3, Issue 7, July 2013. [4] P.M.Meshram, Rohit G. Kanojiya, “Tuning of PID Controller using ziegler-nichols method for Speed Control of DC motor”,IEEE international confrence on advanced inengineering, science and management (ICAESM-2012) march 30,31,2012. [5] Wen-Jun Xu, “Permanent Magnet Synchronous Motor with Linear Quadratic Speed Controller”, 2011 2nd International Conference on Advances in Energy Engineering (ICAEE 2011), Elsevier ,Wen-JuWn eXnu-/J uEnn eXrug\y / P Ernoecregdyi aP 0ro0c (e2d0i1a1 1) 40 0(200–1020)0 3 64 – 369. [6] Aamir Hashim Obeid Ahmed, “ Optimal Speed Control for Direct Current Motors Using Linear Quadratic Regulator”, Journal of Science and Technology vol. 13 ISSN 1605 – 427X Engineering and Computer Sciences (E C S No. 2). [7] https://www.wikipedia.org/ [8] Safina Al Nisa, Lini Mathew, S Chatterji, “Comparative Analysis of Speed Control of DC Motor Using AI Technique”, International Journal of Engineering Research and Applications (IJERA), Vol. 3, Issue 3, May-Jun 2013, pp.1137-1146. [9]S.Z.Moussavi1,M.Alasvandi2,Sh.Javadi3, “A Novel Combination PID Strategy of Speed Control for Independent Excited DC Motor” , Advances in Power and Energy Systems, ISBN: 978-1-61804-128-9. [10] Gwo-Ruey Yu and Rey-Chue Hwang, “Optimal PID Speed Control of Brushless DC Motors Using LQR Approach*”, 2004 IEEE International Conference on Systems, Man and Cybernetics, 0-7803-8566-7/04/2004 IEEE. [11] Aamir Hashim Obeid Ahmed, “ Optimal Speed Control for Direct Current Motors Using Linear Quadratic Regulator”, Journal of Science and Technology - Engineering and Computer Sciences, Vol. 14, No. 3, June 2013. [12] Ankit Rastogi, Pratibha Tiwari, “Optimal Tuning of Fractional Order PID Controller for DC Motor Speed Control Using Particle Swarm Optimization”, International Journal of Soft Computing and Engineering (IJSCE) ISSN: 2231-2307, Volume-3, Issue-2, May 2013.. Vivek Shrivastva, IJRIT. 348.

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